I Proofs
1 Propositions 5
1.1 Compound Propositions 6
1.2 Propositional Logic in Computer Programs 10
1.3 Predicates and Quantifiers 11
1.4 Validity 19
1.5 Satisfiability 21
2 Patterns of Proof 23
2.1 The Axiomatic Method 23
2.2 Proof by Cases 26
2.3 Proving an Implication 27
2.4 Proving an “If and Only If” 30
2.5 Proof by Contradiction 32
2.6 Proofs about Sets 33
2.7 Good Proofs in Practice 40
3 Induction 43
3.1 The Well Ordering Principle 43
3.2 Ordinary Induction 46
3.3 Invariants 56
3.4 Strong Induction 64
3.5 Structural Induction 69
4 Number Theory 81
4.1 Divisibility 81
4.2 The Greatest Common Divisor 87
4.3 The Fundamental Theorem of Arithmetic 94
4.4 Alan Turing 96
4.5 Modular Arithmetic 100
4.6 Arithmetic with a Prime Modulus 103
4.7 Arithmetic with an Arbitrary Modulus 108
4.8 The RSA Algorithm 113
II Structures
5 Graph Theory 121
5.1 Definitions 121
5.2 Matching Problems 128
5.3 Coloring 143
5.4 Getting from A to B in a Graph 147
5.5 Connectivity 151
5.6 Around and Around We Go 156
5.7 Trees 162
5.8 Planar Graphs 170
6 Directed Graphs 189
6.1 Definitions 189
6.2 Tournament Graphs 192
6.3 Communication Networks 196
7 Relations and Partial Orders 213
7.1 Binary Relations 213
7.2 Relations and Cardinality 217
7.3 Relations on One Set 220
7.4 Equivalence Relations 222
7.5 Partial Orders 225
7.6 Posets and DAGs 226
7.7 Topological Sort 229
7.8 Parallel Task Scheduling 232
7.9 Dilworth’s Lemma 235
8 State Machines 237
III Counting
9 Sums and Asymptotics 243
9.1 The Value of an Annuity 244
9.2 Power Sums 250
9.3 Approximating Sums 252
9.4 Hanging Out Over the Edge 257
9.5 Double Trouble 269
9.6 Products 272
9.7 Asymptotic Notation 275
10 Recurrences 283
10.1 The Towers of Hanoi 284
10.2 Merge Sort 291
10.3 Linear Recurrences 294
10.4 Divide-and-Conquer Recurrences 302
10.5 A Feel for Recurrences 309
11 Cardinality Rules 313
11.1 Counting One Thing by Counting Another 313
11.2 Counting Sequences 314
11.3 The Generalized Product Rule 317
11.4 The Division Rule 321
11.5 Counting Subsets 324
11.6 Sequences with Repetitions 326
11.7 Counting Practice: Poker Hands 329
11.8 Inclusion-Exclusion 334
11.9 Combinatorial Proofs 339
11.10 The Pigeonhole Principle 342
11.11 A Magic Trick 346
12 Generating Functions 355
12.1 Definitions and Examples 355
12.2 Operations on Generating Functions 356
12.3 Evaluating Sums 361
12.4 Extracting Coefficients 363
12.5 Solving Linear Recurrences 370
12.6 Counting with Generating Functions 374
13 Infinite Sets 379
13.1 Injections, Surjections, and Bijections 379
13.2 Countable Sets 381
13.3 Power Sets Are Strictly Bigger 384
13.4 Infinities in Computer Science 386
IV Probability
14 Events and Probability Spaces 391
14.1 Let’s Make a Deal 391
14.2 The Four Step Method 392
14.3 Strange Dice 402
14.4 Set Theory and Probability 411
14.5 Infinite Probability Spaces 413
15 Conditional Probability 417
15.1 Definition 417
15.2 Using the Four-Step Method to Determine Conditional Probability 418
15.3 A Posteriori Probabilities 424
15.4 Conditional Identities 427
16 Independence 431
16.1 Definitions 431
16.2 Independence Is an Assumption 432
16.3 Mutual Independence 433
16.4 Pairwise Independence 435
16.5 The Birthday Paradox 438
17 Random Variables and Distributions 445
17.1 Definitions and Examples 445
17.2 Distribution Functions 450
17.3 Bernoulli Distributions 452
17.4 Uniform Distributions 453
17.5 Binomial Distributions 456
18 Expectation 467
18.1 Definitions and Examples 467
18.2 Expected Returns in Gambling Games 477
18.3 Expectations of Sums 483
18.4 Expectations of Products 490
18.5 Expectations of Quotients 492
19 Deviations 497
19.1 Variance 497
19.2 Markov’s Theorem 507
19.3 Chebyshev’s Theorem 513
19.4 Bounds for Sums of Random Variables 516
19.5 Mutually Independent Events 523
20 Random Walks 533
20.1 Unbiased Random Walks 533
20.2 Gambler’s Ruin 542
20.3 Walking in Circles 549
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